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Certificates of optimality for minimum norm biproportional apportionments. (English) Zbl 1314.91186

Summary: Computing a biproportional apportionment that satisfies some given properties may require a high degree of mathematical expertise, that very few voters can share. It seems therefore that the voters have to accept the electoral outcome without any possibility of checking the validity of the stated properties. However, it is possible in some cases to attach to the computed apportionment a certificate which can guarantee the voters of the validity of the apportionment. This type of investigation has been first proposed in [P. Serafini and B. Simeone, ibid. 38, No. 2, 247–268 (2012; Zbl 1244.91033)]. In this paper, we pursue the same line of approach and show that a certificate can be produced and easily checked by a layman for apportionments that minimize either an \(L_1\)- or an \(L_2\)-norm deviation from given quotas.

MSC:

91F10 History, political science
91B32 Resource and cost allocation (including fair division, apportionment, etc.)

Citations:

Zbl 1244.91033

Software:

Bazi
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balinski M, Demange G (1989) An axiomatic approach to proportionality between matrices. Math Oper Res 14:700-719 · Zbl 0689.15001 · doi:10.1287/moor.14.4.700
[2] Balinski M, Demange G (1989) Algorithms for proportional matrices in reals and integers. Math Program 45:193-210 · Zbl 0681.90087 · doi:10.1007/BF01589103
[3] Balinski ML, Ramírez V (1997) Mexican electoral law: 1996 version. Elect Studies 16:329-349 · doi:10.1016/S0261-3794(97)00025-5
[4] Cox LH, Ernst LR (1982) Controlled rounding. INFOR-Info Sys Oper Res 20:423-432 · Zbl 0501.90065
[5] Gaffke N, Pukelsheim F (2008) Divisor methods for proportional representation systems: an optimization approach to vector and matrix apportionment problems. Math Soc Sci 56:166-184 · Zbl 1143.49030 · doi:10.1016/j.mathsocsci.2008.01.004
[6] Gaffke N, Pukelsheim F (2008) Vector and matrix apportionment problems and separable convex integer optimization. Math Methods Oper Res 67:133-159 · Zbl 1152.90537 · doi:10.1007/s00186-007-0184-7
[7] Hillier FS, Liebermann GJ (2010) Introduction to operations research. Mc Graw Hill, New York
[8] Maier S, Pukelsheim F (207) Bazi: a free computer program for proportional representation apportionment Preprint Nr. 042/2007. Institut für Mathematik, Universität Augsburg. http://www.opus-bayern.de/uni-augsburg/volltexte/2007/711/
[9] Kalantari B, Lari I, Ricca F, Simeone B (2008) On the complexity of general matrix scaling and entropy minimization via the ras algorithm. Math Program Ser A112:371-401 · Zbl 1191.15005
[10] Pennisi A (2006) In: Simeone B, Pukelsheimpp F (eds) Mathematics and democracy: recent advances in voting systems and collective choice. The Italian bug: a flawed procedure for bi-proportional seat allocation. Springer, Berlin
[11] Pennisi A, Ricca F, Simeone B (2005a) Malfunzionamenti dell’allocazione biproporzionale di seggi nella riforma elettorale italiana. Dipartimento di Statistica, Probabilità e Statistiche Applicate, Serie A - Ricerche, Università La Sapienza, Roma · Zbl 0501.90065
[12] Pennisi A, Ricca F, Simeone B (2005b) Legge elettorale con paradosso. La Voce, 11 Novembre
[13] Pennisi A, Ricca F, Simeone B (2006) Bachi e buchi della legge elettorale italiana nell’allocazione biproporzionale di seggi. Sociologia e Ricerca Sociale 79:55-76
[14] Pukelsheim F (2004) BAZI: a Java program for proportional representation. Oberwolfach Reports 1: 735-737. http://www.uni-augsburg.de/bazi · Zbl 0689.15001
[15] Pukelsheim F, Ricca F, Scozzari A, Serafini P, Simeone B (2012) Network flow methods for electoral systems. Networks 59:73-88. doi:10.1002/net.20480 · Zbl 1241.91043 · doi:10.1002/net.20480
[16] Ricca F, Scozzari A, Serafini P, Simeone B (2012) Error minimization methods in biproportional apportionment. TOP 20:547-577 · Zbl 1262.90204 · doi:10.1007/s11750-012-0252-x
[17] Serafini P, Simeone B (2012a) Parametric maximum flow methods for minimax approximation of target quotas in biproportional apportionment. Networks 59:191-208 · Zbl 1241.91044
[18] Serafini P, Simeone B (2012b) Certificates of optimality: the third way to biproportional apportionment. Soc Choice Welf 38:247-268 · Zbl 1244.91033
[19] Simeone B, Pukelsheim F (eds) (2007) Mathematics and democracy: recent advances in voting systems and collective choicestudies in choice and welfare. Springer, Berlin
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